Define the number

Let $x = 1.242424...$

$x = 1.\overline{24}$

Multiply to shift the decimal

Since two digits are repeating, multiply by $100$ to shift the decimal point past the repeating block.

$100x = 124.2424...$

Subtract the original equation

Subtract the equation $x = 1.2424...$ from $100x = 124.2424...$.

$100x - x = 124.2424... - 1.2424...$

Simplify

Perform the subtraction to solve for $x$.

$99x = 123 \implies x = \frac{123}{99}$

Reduce the fraction

Divide the numerator and denominator by their greatest common divisor, which is 3.

$x = \frac{123 \div 3}{99 \div 3} = \frac{41}{33}$